Ellipse Involute

$\begin{figure}\begin{center}\BoxedEPSF{EllipseInvolute.epsf scaled 800}\end{center}\end{figure}$

$\begin{displaymath} {\bf T}=\left[{\matrix{-a\sin t\cr b\cos t}}\right], \end{displaymath}$

(1)

and the Arc Length is

$\begin{displaymath} s=a\int\sqrt{1-e^2\sin^2 t}\,dt = aE(t,e), \end{displaymath}$

(2)

where

$\displaystyle {\bf r}_i$	$\textstyle =$	$\displaystyle {\bf r}-s\hat{\bf T}=\left[\begin{array}{c}a\cos t\\ b\sin t\end{array}\right]-aeE(t,e)\left[\begin{array}{c}-a\sin t\\ b\cos t\end{array}\right]$	(3)
	$\textstyle =$	$\displaystyle \left[\begin{array}{c}a\{\cos t+aeE(t,e)\sin t\}\\ b\{\sin t-aeE(t,e)\cos t\}.\end{array}\right]$	(4)